A Generalized Lyapunov-Sylvester Computational Method for Numerical Solutions of NLS Equation with Singular Potential

In the present paper a numerical method is developed to approximate the solution of two-dimensional Nonlinear Schrodinger equation in the presence of a sin- gular potential. The method leads to generalized Lyapunov-Sylvester algebraic opera- tors that are shown to be invertible using original topological and differential calculus issued methods. The numerical scheme is proved to be consistent, convergent and sta- ble using the Lyapunov criterion, lax equivalence theorem and the properties of the generalized Lyapunov-Sylvester operators.

Improved gradient iterative algorithms for solving Lyapunov matrix equations

In this paper, an improved gradient iterative （GI） algorithm for solving the Lyapunov matrix equations is studied. Convergence of the improved method for any initial value is proved with some conditions. Compared with the GI algorithm, the improved algorithm reduces computational cost and storage. Finally, the algorithm is tested with GI several numerical examples.

AOR Iterative Method for Coupled Lyapunov Matrix Equations

An AOR(Accelerated Over-Relaxation)iterative method is suggested by introducing one more parameter than SOR(Successive Over-Relaxation)method for solving coupled Lyapunov matrix equations(CLMEs)that come from continuous-time Markovian jump linear systems.The proposed algorithm improves the convergence rate,which can be seen from the given illustrative examples.The comprehensive theoretical analysis of convergence and optimal parameter needs further investigation.

Solution of Algebraic Lyapunov Equation on Positive-Definite Hermitian Matrices by Using Extended Hamiltonian Algorithm

This communique is opted to study the approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices.We choose the geodesic distance between􀀀AHX􀀀XA and P as the cost function,and put forward the Extended Hamiltonian algorithm(EHA)and Natural gradient algorithm(NGA)for the solution.Finally,several numerical experiments give you an idea about the effectiveness of the proposed algorithms.We also show the comparison between these two algorithms EHA and NGA.Obtained results are provided and analyzed graphically.We also conclude that the extended Hamiltonian algorithm has better convergence speed than the natural gradient algorithm,whereas the trajectory of the solution matrix is optimal in case of Natural gradient algorithm(NGA)as compared to Extended Hamiltonian Algorithm(EHA).The aim of this paper is to show that the Extended Hamiltonian algorithm(EHA)has superior convergence properties as compared to Natural gradient algorithm(NGA).Upto the best of author’s knowledge,no approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices is found so far in the literature.

Lyapunov-type Inequalities for a System of Nonlinear Differential Equations

This paper presents several new Lyapunov-type inequalities for a system of first-order nonlinear differential equations. Our results generalize and improve some existing ones.

On the Measurement of Lower Solution Bounds of the Discrete Algebraic Lyapunov Equation

In this paper, estimations of the lower solution bounds for the discrete algebraic Lyapunov Equation (the DALE) are addressed. By utilizing linear algebraic techniques, several new lower solution bounds of the DALE are presented. We also propose numerical algorithms to develop sharper solution bounds. The obtained bounds can give a supplement to those appeared in the literature.

四元数Lyapunov方程的酉结构解及最佳逼近

针对四元数Lyapunov方程AX+XA*=C,在A、C为正规矩阵条件下,利用四元数矩阵的Frobenius范数酉乘积不变性和矩阵的右特征值分解,得到了该方程存在酉结构解的充分必要条件及其通解表达式。同时对预先给定的酉矩阵M,应用四元数矩阵的迹不等式和矩阵的分块方法,在该方程的酉结构解集中得到与M的最佳逼近解。最后给出求解步骤,通过数值算例和四元数矩阵的复表示运算,检验了所得理论结果的正确性及所给方法的可行性。

A New Seven-modes Truncation of the Plane Incompressible Navier-Stokes EquationsA New Seven-modes Truncation of the Plane Incompressible Navier-Stokes Equations

A new seven-modes truncation of Fourier series of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained.And its stationary solutions,the existence of attractor and the global stability of the equations are firmly proved.At the same time,several issues such as some basic dynamical behaviors and routs to chaos are shown numerically by changing Reynolds number.The system exhibits a stochastic behavior approached through an involved sequence of bifurcations.

NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE:MOMENTS AND INTERMITTENCY

In this article, we study the nonlinear stochastic heat equation in the spatial domain R^d subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang's condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on Z^d to that on R^d.Then, we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan [9] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al [17] using different techniques.

Asymptotic stability for impulsive functional differential equations

In this paper, we investigate the stability of a class of impulsive functional differential equations by using Lyapunov functional and Jensen＇s inequality. Some new stability theorems are obtained. Examples are given to demonstrate the advantage of the obtained results.

Optimal control approach to stability criteria on Hill's equations

This paper gives a tutorial on how to prove Lyapunov type criteria by optimal control methods. Firstly, we consider stability criteria on Hill’s equations with nonnegative potential. By optimal control methods developed in 1990s, we obtain several stability criteria including Lyapunov’s criterion, Neǐgauz and Lidskiǐ’s criterion. Secondly, we present stability criteria on Hill’s equations with sign-changing potential in which Brog’s criterion and Krein’s criterion are included.

Stability of Global Solution to Boltzmann-Enskog Equation with External Force

In the presence of external forces depending only on the time and space variables, the Boltzmann-Enskog equation formally conserves only the mass of the . system, and its entropy functional is also nonincreasing. Corresponding to this type of equation, we first give some hypotheses of its bicharacteristic equations and then get some results about the stablity of its global solution with the help of two new Lyapunov functionals： one is to describe interactions between particles with different velocities and the other is to measure the L1 distance between two mild solutions. The former Lyapunov functional yields the time-asymptotic convergence of global classical solutions to the collision free motion while the latter is applied into the veri-fication of the L1 stability of global mild solutions to the Boltzmann-Enskog equation for a moderately or highly dense gas in the influence of external forces.

On the Stability and Boundedness of Solutions of Certain Non-Autonomous Delay Differential Equation of Third Order

In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the equations. The conditions stated complement previously known results. Example is also given to illustrate the correctness and significance of the result obtained.

Comparison on Sufficient Conditions for the Stability of Hill Equation: An Arnold’s Tongues Approach

It is known that the solutions of a second order linear differential equation with periodic coefficients are almost always analytically impossible to obtain and in order to study its properties we often require a computational approach. In this paper we compare graphically, using the Arnold Tongues, some sufficient criteria for the stability of periodic differential equations. We also present a brief explanation on how the authors, of each criterion, obtained them. And a comparison between four sufficient stability criteria and the stability zones found by perturbation methods is presented.

A Nine-modes Truncation of the Plane Incompressible Navier-Stokes Equations

In this paper a nine-modes truncation of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained. The stationary solutions, the existence of attractor and the global stability of the equations are firmly proved. What is more, that the force f acts on the mode ks and k7 respectively produces two systems, which lead to a much richer and varied phenomenon. Numerical simulation is given at last, which shows a stochastic behavior approached through an involved sequence of bifurcations.

On Stability of Solutions of a Certain Fourth-Order Functional Differential Equations

Using a Razumikhin-type theorem,we obtain sufficient conditions for the global asymptotic stability of the zero solution of a certain fourth order functional differential equations.The result generalizes the well known results.

A Five-Mode System of the Navier-Stokes Equations on a Torus

A five-mode truncation of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is studied. Its stationary solutions and stability are presented;the existence of attractor and the global stability of the system are discussed. The whole process, which shows a chaos behavior approached through an involved sequence of bifurcations with the changing of Reynolds number, is simulated numerically. Based on numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum, Poincare section, power spectrum and return map of the system, some basic dynamical behavior of the new chaos system are revealed.

An Iterative Relaxation Approach to the Solution of the Hamilton-Jacobi-Bellman-Isaacs Equation in Nonlinear Optimal Control

In this paper, we propose an iterative relaxation method for solving the Hamilton-Jacobi-Bellman-Isaacs equation(HJBIE) arising in deterministic optimal control of affine nonlinear systems. Local convergence of the method is established under fairly mild assumptions, and examples are solved to demonstrate the effectiveness of the method. An extension of the approach to Lyapunov equations is also discussed. The preliminary results presented are promising, and it is hoped that the approach will ultimately develop into an efficient computational tool for solving the HJBIEs.

An iterative algorithm for solving a class of matrix equations

In this paper, an iterative algorithm is presented to solve the Sylvester and Lyapunov matrix equations. By this iterative algorithm, for any initial matrix X1, a solution X＊ can be obtained within finite iteration steps in the absence of roundoff errors. Some examples illustrate that this algorithm is very efficient and better than that of [ 1 ] and [2].